First: I know that there is some questions on MSE that cover this problem, but I didn't find any answer that solves the problem straightforward by the method of the primitive element theorem.
I want to find a primitive element for the extension $K | \Bbb{Q}$, where $K$ is the splitting field of $f(X)=X^4-2$ over $\Bbb{Q}.$
I know that $K=\Bbb{Q}(\sqrt[4]{2},i\sqrt[4]{2})=\Bbb{Q}(\sqrt[4]{2},i)$. If $\alpha=\sqrt[4]{2}$ and $\beta=i$, I know that exists an primitive element $\gamma=\alpha+c\beta$, $c\in \Bbb{Q}$ such that $K=\Bbb{Q}(\gamma),$ so I just need to find such $c$. My question is to how to it.
I know that $[K,\Bbb{Q}]=8$, so I tried to find $c\in\Bbb{Q}$ such that $\gamma\in K$ and $\deg (p_{\gamma})=8$, where $p_{\gamma}$ si the minimal polynomial of $\gamma$ over $\Bbb{Q}$. If I do it, so I can take $\gamma$ as primitive element.
However, I coldn't and I don't know if there is some straightforward method to solve this problem.
Thank you.